Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $n \neq 0$. $p = \dfrac{-4n + 40}{4n + 40} \div \dfrac{-4n^2 + 60n - 200}{4n^2 - 36n + 80} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{-4n + 40}{4n + 40} \times \dfrac{4n^2 - 36n + 80}{-4n^2 + 60n - 200} $ First factor out any common factors. $p = \dfrac{-4(n - 10)}{4(n + 10)} \times \dfrac{4(n^2 - 9n + 20)}{-4(n^2 - 15n + 50)} $ Then factor the quadratic expressions. $p = \dfrac {-4(n - 10)} {4(n + 10)} \times \dfrac {4(n - 5)(n - 4)} {-4(n - 5)(n - 10)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac {-4(n - 10) \times 4(n - 5)(n - 4) } {4(n + 10) \times -4(n - 5)(n - 10) } $ $p = \dfrac {-16(n - 5)(n - 4)(n - 10)} {-16(n - 5)(n - 10)(n + 10)} $ Notice that $(n - 5)$ and $(n - 10)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac {-16\cancel{(n - 5)}(n - 4)(n - 10)} {-16\cancel{(n - 5)}(n - 10)(n + 10)} $ We are dividing by $n - 5$ , so $n - 5 \neq 0$ Therefore, $n \neq 5$ $p = \dfrac {-16\cancel{(n - 5)}(n - 4)\cancel{(n - 10)}} {-16\cancel{(n - 5)}\cancel{(n - 10)}(n + 10)} $ We are dividing by $n - 10$ , so $n - 10 \neq 0$ Therefore, $n \neq 10$ $p = \dfrac {-16(n - 4)} {-16(n + 10)} $ $ p = \dfrac{n - 4}{n + 10}; n \neq 5; n \neq 10 $